Distinguishing number and distinguishing index of some operations on graphs

TL;DR

This paper investigates how the distinguishing number and index of a graph change under various graph operations such as vertex removal, edge removal, and contractions, providing bounds and inequalities for these changes.

Contribution

It establishes new bounds on the changes in the distinguishing number and index when applying common graph operations, extending understanding of graph symmetries.

Findings

Vertex removal changes are bounded between -1 and D(G).

Edge removal alters D(G) and D'(G) by at most 2 and 2 respectively.

Contractions can decrease D(G) and D'(G) by at most 1, or increase them by up to 3 times their original values.

Abstract

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. We examine the effects on $D(G)$ and $D'(G)$ when $G$ is modified by operations on vertex and edge of $G$. Let $G$ be a connected graph of order $n\geq 3$. We show that $-1\leq D(G-v)-D(G)\leq D(G)$, where $G-v$ denotes the graph obtained from $G$ by removal of a vertex $v$ and all edges incident to $v$ and these inequalities are true for the distinguishing index. Also we prove that $|D(G-e)-D(G)|\leq 2$ and $-1 \leq D'(G-e)-D'(G)\leq 2$, where $G-e$ denotes the graph obtained from $G$ by simply removing the edge $e$. Finally we consider the vertex contraction and the edge contraction of $G$ and prove that the edge contraction decrease the distinguishing number (index) of $G$ by at most one and increase by at most $3D(G)$ ($3D'(G)$).